arXiv subjects

Peter Grünwald

Publications and source records attributed to Peter Grünwald.

At least 19 recordsLinked to original sources

arXiv2026

E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values

A recurring debate in the philosophy of statistics concerns what, exactly, should count as a measure of evidence for or against a given hypothesis. P-values, likelihood ratios, and Bayes factors all have their defenders. In this paper we add two additional candidates to this list: the e-value and its sequential analogue, the e-process. E-values enjoy several desirable properties as measures of evidence: they combine naturally across studies, handle composite hypotheses, provide long-run error rates, and admit a useful interpretation as the wealth accrued by a bettor in a game against the null distribution. E-processes additionally handle optional stopping and optional continuation. This work examines the extent to which e-values and e-processes satisfy the evidential desiderata of different statistical traditions, concluding that they combine attractive features of p-values, likelihood ratios, and Bayes factors, and merit serious consideration as interpretable and intuitive measures of statistical evidence.

stat.ME↗

arXiv2026

Interaction between Rydberg Excitons in Cuprous Oxide Revealed through Resonant Second Harmonic Generation

We report experimental and theoretical investigations of interacting excitons of the yellow series in cuprous oxide (Cu$_2$O) with principal quantum numbers up to by means of second harmonic generation (SHG). Using picosecond pulsed laser excitation up to 10 GW/cm$^2$ peak intensity we observe a pronounced change of the spectra with increasing pump laser intensity: an energetic shift to lower absolute energies and a spectral broadening. The absolute intensities of the spectral lines scale for low powers with the square of the pump power, but saturates at higher powers. At still higher powers the SHG intensity is actually reduced. To explain these results quantitively, we developed a semi-classical theory of resonant SHG where the process of SHG is fully coherent. The excitons are assumed to be bosons interacting by a distance dependent potential giving rise to both the changes in spectral line shape and the saturation by a $\it{Rydberg}$ blockade. The concomitant measurement of two-photon absorption allows to derive quantitative values for the exciton-exciton interaction. While the results agree in order of magnitude with those calculated by state-of-the art atomic-like van der Waals interaction theory, the scaling with principle quantum number is quite different. As a possible screening by an electron-hole plasma created by three-photon absorption into blue and violet band states could be ruled out, our results point toward fundamental differences between excitons and atoms.

cond-mat.mes-hall↗

arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2026

On admissibility in post-hoc hypothesis testing

The validity of classical hypothesis testing requires the significance level $α$ be fixed before any statistical analysis takes place. This is a stringent requirement. For instance, it prohibits updating $α$ during (or after) an experiment due to changing concern about the cost of false positives, or to reflect unexpectedly strong evidence against the null. Perhaps most disturbingly, witnessing a p-value $p\llα$ vs $p= α- ε$ for tiny $ε> 0$ has no (statistical) relevance for any downstream decision-making. Following recent work of Grünwald (2024), we develop a theory of post-hoc hypothesis testing, enabling $α$ to be chosen after seeing and analyzing the data. To study "good" post-hoc tests we introduce $Γ$-admissibility, where $Γ$ is a set of adversaries which map the data to a significance level. We classify the set of $Γ$-admissible rules for various sets $Γ$, showing they must be based on e-values, and recover the Neyman-Pearson lemma when $Γ$ is the constant map.

math.ST↗

arXiv2025

Rao-Blackwellized e-variables

We show that for any concave utility, the expected utility of an e-variable can only increase after conditioning on a sufficient statistic. The simplest form of the result has an extremely straightforward proof, which follows from a single application of Jensen's inequality. Similar statements hold for compound e-variables, asymptotic e-variables, and e-processes. These results echo the Rao-Blackwell theorem, which states that the expected squared error of an estimator can only decrease after conditioning on a sufficient statistic. We provide several applications of this insight, including a simplified derivation of the log-optimal e-variable for linear regression with known variance.

math.ST↗

arXiv2025

Testing maximum entropy models with e-values

E-values have recently emerged as a robust and flexible alternative to p-values for hypothesis testing, especially under optional continuation, i.e., when additional data from further experiments are collected. In this work, we define optimal e-values for testing between maximum entropy models, both in the microcanonical (hard constraints) and canonical (soft constraints) settings. We show that, when testing between two hypotheses that are both microcanonical, the so-called growth-rate optimal e-variable admits an exact analytical expression, which also serves as a valid e-variable in the canonical case. For canonical tests, where exact solutions are typically unavailable, we introduce a microcanonical approximation and verify its excellent performance via both theoretical arguments and numerical simulations. We then consider constrained binary models, focusing on $2 \times k$ contingency tables -- an essential framework in statistics and a natural representation for various models of complex systems. Our microcanonical optimal e-variable performs well in both settings, constituting a new tool that remains effective even in the challenging case when the number $k$ of groups grows with the sample size, as in models with growing features used for the analysis of real-world heterogeneous networks and time-series.

stat.ME↗

arXiv2025

E-Values for Exponential Families: the General Case

We analyze common types of e-variables and e-processes for composite exponential family nulls: the optimal e-variable based on the reverse information projection (RIPr), the conditional (COND) e-variable, and the universal inference (UI) and sequen\-tialized RIPr e-processes. We characterize the RIPr prior for simple and Bayes-mixture based alternatives, either precisely (for Gaussian nulls and alternatives) or in an approximate sense (general exponential families). We provide conditions under which the RIPr e-variable is (again exactly vs. approximately) equal to the COND e-variable. Based on these and other interrelations which we establish, we determine the e-power of the four e-statistics as a function of sample size, exactly for Gaussian and up to $o(1)$ in general. For $d$-dimensional null and alternative, the e-power of UI tends to be smaller by a term of $(d/2) \log n + O(1)$ than that of the COND e-variable, which is the clear winner.

stat.ME↗

arXiv2025

Description length of canonical and microcanonical models

The (non-)equivalence of canonical and microcanonical ensembles is a fundamental question in statistical physics, concerning whether the use of soft and hard constraints in the maximum-entropy construction leads to the same description of a system. Despite the fact that maximum-entropy models are also commonly used in statistical inference, pattern detection, and hypothesis testing, a complete understanding of the effects of ensemble non-equivalence on statistical modeling is still missing. Here, we study this problem from a rigorous model selection perspective by comparing canonical and microcanonical models via the Minimum Description Length (MDL) principle, which yields a trade-off between likelihood, measuring model accuracy, and complexity, measuring model flexibility and its potential to overfit data. We compute the Normalized Maximum Likelihood (NML) of both formulations and find that: (i) microcanonical models always achieve higher likelihood but are always more complex; (ii) the optimal model choice depends on the empirical values of the constraints -- the canonical model performs best when its fit to the observed data exceeds its uniform average fit across all realizations; (iii) in the thermodynamic limit, the difference in description length per node vanishes when ensemble equivalence holds but persists otherwise, showing that non-equivalence implies extensive differences between large canonical and microcanonical models. Finally, we compare the NML approach to Bayesian methods, showing that (iv) the choice of priors, practically irrelevant in equivalent models, becomes crucial when an extensive number of constraints is enforced, possibly leading to very different outcomes.

cond-mat.stat-mech↗

arXiv2025

Interaction between Rydberg Excitons in Cuprous Oxide Revealed through Second Harmonic Generation

We report experimental and theoretical investigations of interacting excitons of the yellow series in cuprous oxide (Cu$_2$O) with principal quantum numbers up to $n=7$ by means of second harmonic generation (SHG). Using picosecond pulsed laser excitation up to 10 GW/cm$^2$ peak intensity we observe a pronounced change of the spectra with increasing pump laser intensity: an energetic shift to lower absolute energies and a spectral broadening, while the absolute intensity for low powers scales with the square of the pump power, but saturates at higher powers. Concomitant with SHG we determined the density of the excitons excited by the ps pulse by measuring the two-photon absorption directly. This allows to derive quantitative values for the exciton-exciton interaction. Surprisingly, the results disagree both in magnitude and scaling with principal quantum number with those calculated by state-of-the art atomic-like van der Waals interaction theory. As a possible screening by an electron-hole plasma created by three-photon absorption into blue and violet band states could be ruled out, our results point toward fundamental differences between excitons and atoms.

cond-mat.mtrl-sci↗

arXiv2025

ALL-IN meta-analysis: breathing life into living systematic reviews and prospective meta-analyses

Science is justly admired as a cumulative process ("standing on the shoulders of giants"), yet scientific knowledge is typically built on a patchwork of research contributions without much coordination. This lack of efficiency has specifically been addressed in clinical research by recommendations against avoidable research waste and for living systematic reviews and prospective meta-analysis. We propose to further those recommendations with ALL-IN meta-analysis: Anytime Live and Leading INterim meta-analysis. ALL-IN provides meta-analysis based on e-values and anytime-valid confidence intervals that can be updated at any time - reanalyzing after each new observation while retaining type-I error and coverage guarantees, live - no need to prespecify the looks, and leading - in the decisions on whether individual studies should be initiated, stopped or expanded, the meta-analysis can be the leading source of information without losing validity to accumulation bias. The analysis design requires no information about the trial sample sizes or the number of trials eventually included. So ALL-IN meta-analysis can be applied retrospectively as well as prospectively, to evaluate the evidence once or sequentially. Because the intention of the analysis does not change the validity of the results, the results of the analysis can change the intentions ('optional stopping' and 'optional continuation' based on the results so far). On the one hand: any analysis can be turned into a living one, or even become prospective and real-time by updating with new trial data and including interim data from trials that are still ongoing - without any changes in the cut-offs for testing or the method for interval estimation. On the other hand: no stopping rule needs to be enforced for the analysis to remain valid, so a prospective meta-analysis can be a bottom-up collaboration [...]

stat.ME↗

arXiv2025

Optimal E-Values for Exponential Families: the Simple Case

We provide a general condition under which e-variables in the form of a simple-vs.-simple likelihood ratio exist when the null hypothesis is a composite, multivariate exponential family. Such `simple' e-variables are easy to compute and expected-log-optimal with respect to any stopping time. Simple e-variables were previously only known to exist in quite specific settings, but we offer a unifying theorem on their existence for testing exponential families. We start with a simple alternative $Q$ and a regular exponential family null. Together these induce a second exponential family ${\cal Q}$ containing $Q$, with the same sufficient statistic as the null. Our theorem shows that simple e-variables exist whenever the covariance matrices of ${\cal Q}$ and the null are in a certain relation. A prime example in which this relation holds is testing whether a parameter in a linear regression is 0. Other examples include some $k$-sample tests, Gaussian location- and scale tests, and tests for more general classes of natural exponential families. While in all these examples, the implicit composite alternative is also an exponential family, in general this is not required.

stat.ME↗

arXiv2024

Growth-Optimal E-Variables and an extension to the multivariate Csiszár-Sanov-Chernoff Theorem

We consider growth-optimal e-variables with maximal e-power, both in an absolute and relative sense, for simple null hypotheses for a $d$-dimensional random vector, and multivariate composite alternatives represented as a set of $d$-dimensional means $\meanspace_1$. These include, among others, the set of all distributions with mean in $\meanspace_1$, and the exponential family generated by the null restricted to means in $\meanspace_1$. We show how these optimal e-variables are related to Csiszár-Sanov-Chernoff bounds, first for the case that $\meanspace_1$ is convex (these results are not new; we merely reformulate them) and then for the case that $\meanspace_1$ `surrounds' the null hypothesis (these results are new).

cs.IT↗

arXiv2024

Reverse Information Projections and Optimal E-statistics

Information projections have found important applications in probability theory, statistics, and related areas. In the field of hypothesis testing in particular, the reverse information projection (RIPr) has recently been shown to lead to growth-rate optimal (GRO) e-statistics for testing simple alternatives against composite null hypotheses. However, the RIPr as well as the GRO criterion are undefined whenever the infimum information divergence between the null and alternative is infinite. We show that in such scenarios, under some assumptions, there still exists a measure in the null that is closest to the alternative in a specific sense. Whenever the information divergence is finite, this measure coincides with the usual RIPr. It therefore gives a natural extension of the RIPr to certain cases where the latter was previously not defined. This extended notion of the RIPr is shown to lead to optimal e-statistics in a sense that is a novel, but natural, extension of the GRO criterion. We also give conditions under which the (extension of the) RIPr is a strict sub-probability measure, as well as conditions under which an approximation of the RIPr leads to approximate e-statistics. For this case we provide tight relations between the corresponding approximation rates.

cs.IT↗

arXiv2024

Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha

A standard practice in statistical hypothesis testing is to mention the p-value alongside the accept/reject decision. We show the advantages of mentioning an e-value instead. With p-values, it is not clear how to use an extreme observation (e.g. p $\ll α$) for getting better frequentist decisions. With e-values it is straightforward, since they provide Type-I risk control in a generalized Neyman-Pearson setting with the decision task (a general loss function) determined post-hoc, after observation of the data -- thereby providing a handle on `roving $α$'s'. When Type-II risks are taken into consideration, the only admissible decision rules in the post-hoc setting turn out to be e-value-based. Similarly, if the loss incurred when specifying a faulty confidence interval is not fixed in advance, standard confidence intervals and distributions may fail whereas e-confidence sets and e-posteriors still provide valid risk guarantees. Sufficiently powerful e-values have by now been developed for a range of classical testing problems. We discuss the main challenges for wider development and deployment.

stat.ME↗

arXiv2024

E-values for k-Sample Tests With Exponential Families

We develop and compare e-variables for testing whether $k$ samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal) e-variables for (1) a `small' null inside the same exponential family, and (2) a `large' nonparametric null, as well as (3) an e-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. [2021] and Wald [1947] respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of these e-variables in terms of their logarithmic growth rate, and find that for small effects all four e-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster under the alternative is family-dependent. We furthermore discuss algorithms for numerically approximating (1).

stat.ME↗

arXiv2023

E-Statistics, Group Invariance and Anytime Valid Testing

We study worst-case-growth-rate-optimal (GROW) e-statistics for hypothesis testing between two group models. It is known that under a mild condition on the action of the underlying group G on the data, there exists a maximally invariant statistic. We show that among all e-statistics, invariant or not, the likelihood ratio of the maximally invariant statistic is GROW, both in the absolute and in the relative sense, and that an anytime-valid test can be based on it. The GROW e-statistic is equal to a Bayes factor with a right Haar prior on G. Our treatment avoids nonuniqueness issues that sometimes arise for such priors in Bayesian contexts. A crucial assumption on the group G is its amenability, a well-known group-theoretical condition, which holds, for instance, in scale-location families. Our results also apply to finite-dimensional linear regression.

math.ST↗

arXiv2023

The E-Posterior

We develop a representation of a decision maker's uncertainty based on e-variables. Like the Bayesian posterior, this *e-posterior* allows for making predictions against arbitrary loss functions that may not be specified ex ante. Unlike the Bayesian posterior, it provides risk bounds that have frequentist validity irrespective of prior adequacy: if the e-collection (which plays a role analogous to the Bayesian prior) is chosen badly, the bounds get loose rather than wrong, making *e-posterior minimax* decision rules safer than Bayesian ones. The resulting *quasi-conditional paradigm* is illustrated by re-interpreting a previous influential partial Bayes-frequentist unification, *Kiefer-Berger-Brown-Wolpert conditional frequentist tests*, in terms of e-posteriors.

math.ST↗

arXiv2023

Minimax risk classifiers with 0-1 loss

Supervised classification techniques use training samples to learn a classification rule with small expected 0-1 loss (error probability). Conventional methods enable tractable learning and provide out-of-sample generalization by using surrogate losses instead of the 0-1 loss and considering specific families of rules (hypothesis classes). This paper presents minimax risk classifiers (MRCs) that minize the worst-case 0-1 loss with respect to uncertainty sets of distributions that can include the underlying distribution, with a tunable confidence. We show that MRCs can provide tight performance guarantees at learning and are strongly universally consistent using feature mappings given by characteristic kernels. The paper also proposes efficient optimization techniques for MRC learning and shows that the methods presented can provide accurate classification together with tight performance guarantees in practice.

stat.ML↗